Hardware Implementation of Large Number-Multiplication by FFT with Modular Arithmetic
Bibliographic record
Abstract
Modular multiplication (MM) for large integers is the foundation of most public-key cryptosystems, specifically RSA, El-Gamal and the elliptic curve cryptosystems. Thus MM algorithms have been studied widely and extensively. Most of works are based on the well known Montgomery multiplication method (MMM) and its variants, which require multiplication in N. Authors have always avoided the fast Fourier transform (FFT) method believing that it is impractical for present system sizes despite its smaller complexity order. In this paper, the authors presented the design and hardware implementation of a FFT-based algorithm using modular arithmetic to efficiently compute very large number multiplications. The algorithm has been implemented in CASM, an intermediate level HDL developed in the laboratory. The target architecture is a FPGA. The algorithm is scalable and can easily be mapped to any operand size. Results show that such algorithm implementation starts to be useful for 4096-bit operands and beyond.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".